Where Does the δ(t) Term Come From in First Order Unit Step Response?

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The δ(t) term in the equation x' + kx = (r/k + c2)δ(t) represents a Dirac delta function, which is used in control theory and signal processing to model instantaneous impulses. This term evaluates to zero for t ≠ 0, but it has a significant effect at t = 0, where it represents an impulse input to the system. The confusion arises from the interpretation of the delta function, which is not a traditional function but rather a distribution that captures the idea of an instantaneous change. The term (r/k + c2) is a scaling factor for the impulse, indicating the strength of the input at that moment. Understanding the role of δ(t) is crucial for analyzing the system's response to step inputs.
charlie.elvers
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I think I mostly understand how this works, and it makes intuitive sense. However, I'm a little bit confused by one step in the proving of this.

On the second page of the attached PDF, there is this statement:

x' + kx = (r/k + c2)δ(t) ... (followed by the cases of t<0 and t>0

What comes after δ(t) is easy for me to follow, but I don't know where this δ(t) term comes from. Does this term evaluate to 0? If so, how? I would think it'd evaluate to (r/k + c2) because the (r/k + c2) is independent of t

Thanks.
 

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Sorry folks. I just read the forum rules, and I think this is probably the wrong forum for such a thread.
 

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